Area of rectangles

What is this skill?

Area is the amount of flat space a shape covers, measured in square units. A rectangle 5 centimeters wide and 3 centimeters tall can be covered by 3 rows of 5 one-centimeter squares, so its area is 15 square centimeters. Third graders start by counting the squares inside a shape, then notice that the rows and columns make an array.

From that, they learn that the area of a rectangle can be found by multiplying the length of the sides: width × height, or length × width. The answer is always given in square units, such as square inches, square feet or square meters, because area is counted in squares, not lengths.

Why it matters

Area connects multiplication to geometry. The array picture of multiplication becomes a rectangle, and the area model is later used to multiply two-digit numbers, to multiply fractions, and in algebra to multiply expressions. Children who understand area as rows of squares have a picture they will use for years.

Area is also practical: working out how much carpet a room needs, how much paint to buy, or how big a garden bed is. It helps children see that the same amount of space can come in different shapes, such as a 2 by 6 rectangle and a 3 by 4 rectangle that both have an area of 12 square units.

Common mistakes

• Adding the side lengths instead of multiplying, which mixes up area with perimeter. Draw the squares inside the rectangle to show that area counts what is inside.
• Forgetting to write square units, such as writing 15 cm instead of 15 square centimeters. Explain that the answer counts squares, not a length.
• Counting squares carelessly, missing or double-counting some in a large shape. Counting rows and multiplying is faster and more accurate.
• Thinking that a longer rectangle must have a bigger area. Compare a long thin rectangle with a squarer one to see that it depends on both sides.

How to help at home

• Use square tiles, sticky notes or crackers to cover a rectangle, then count and check by multiplying.
• Find the area of real things: a placemat, a book cover or a floor tile, measured in inches or centimeters.
• Draw rectangles on grid paper with a given area, such as 12 squares, and see how many different shapes your child can make.
• Ask "Which is bigger?" about two rugs or two tables and work out the areas to check.
• Plan a pretend garden on graph paper: give your child 24 squares of space and ask them to design a rectangular bed, then say its width, height and area.

Worked examples

These are real questions from the Tutorsaurus area of rectangles practice, with the answers worked out.

1. Example 1

   A rectangle is 2 m wide and 2 m tall. What is its area?

   Answer: 4 sq m. Area = width × height = 2 × 2 = 4 square m.

2. Example 2

   A rectangle is 2 ft wide and 8 ft tall. What is its area?

   Answer: 16 sq ft. Area = width × height = 2 × 8 = 16 square ft.

3. Example 3

   A rectangle is 5 in wide and 9 in tall. What is its area?

   Answer: 45 sq in. Area = width × height = 5 × 9 = 45 square in.